Evaluating Improper Integrals Examples 2

We will now be looking at some more examples of evaluating improper integrals. Please check over the Evaluating Improper Integrals Examples 1 first for earlier examples.

Example 1

Evaluate the following integral: $\int_{-\infty}^{\infty} \frac{1}{1 + x^2} \: dx$.

We will first split up our integral to get to improper integrals, namely $\int_{-\infty}^{\infty} \frac{1}{1 + x^2} \: dx = \int_{-\infty}{0} \frac{1}{1 + x^2} \: dx + \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx$. Note that $\frac{1}{1 + x^2}$ is an even function, so we only have to evaluate one of the improper integrals.

(1)

\begin{align} \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx = \lim_{t \to \infty} \int_{0}^{t} \frac{1}{1 + x^2} \: dx \\ \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx = \lim_{t \to \infty} \tan ^{-1} x \biggr \rvert_{0}^{t} \\ \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx = \lim_{t \to \infty} \tan ^{-1} (t) - \tan^{-1} (0) \\ \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx = \lim_{t \to \infty} \tan ^{-1} (t) \\ \int_{0}^{\infty} \frac{1}{1 + x^2} \: dx = \lim_{t \to \infty} \frac{\pi}{2} \end{align}